Problem 10 - College Algebra with Corequisite Support

Question

For the following exercises, use this scenario: Two different telephone carriers offer the following plans that a person is considering. Company A has a monthly fee of $$\$ 20$$ and charges of $$\$ .05 /$$ min for calls. Company \(B\) has a monthly fee of $$\$ 5$$ and charges $$\$ .10 / \mathrm{min}$$ for calls. Find the model of the total cost of Company B's plan, using \(m\) for the minutes.

Step-by-Step Solution

Answer

The total cost model for Company B is \( C = 5 + 0.10m \).

Step 1: Identify the primary components in Company B's plan

Company B has two components in their pricing structure: a monthly fee and a per-minute charge. We need to model these components to understand the total cost.

Step 2: Decide on the variables

In mathematical modeling, we assign variables to the quantities we want to measure. For this scenario, we will use \( m \) to represent the number of minutes used in a month.

Step 3: Write the fixed monthly cost

Company B charges a fixed monthly fee of \$5. This charge does not depend on the number of minutes \( m \) used.

Step 4: Write the variable cost based on usage

Next, we consider the cost related to the duration of calls. Company B charges \$0.10 per minute. Thus, the variable cost for \( m \) minutes is \( 0.10m \).

Step 5: Combine the fixed and variable costs

The total cost \( C \) of Company B's plan can be modeled by adding the fixed monthly cost and the variable cost: \[ C = 5 + 0.10m \] This equation represents the total monthly cost based on the number of minutes \( m \) you use.

Key Concepts

Cost Analysis

Cost analysis is a crucial factor when choosing between service plans. It involves evaluating the total expenses associated with different options to make informed decisions. In the given scenario, the cost analysis would mean analyzing the plans offered by Company A and Company B to see which one aligns better with the consumer's needs.
Both companies have a combination of fixed and variable costs.

• **Fixed Cost**: This is a monthly fee that does not change, regardless of usage. For Company B, this is set at $5.
• **Variable Cost**: This cost depends on the number of minutes (`m`) used. For Company B, it's $0.10 per minute.

An effective cost analysis will factor in these two elements to assess the total expected outlay in different usage scenarios. This enables us to construct a mathematical model, which is a representation of real-world scenarios expressed through equations. Doing so helps consumers predict monthly expenses under different circumstances and make economically sound choices.

Linear Equations

Linear equations are fundamental tools in mathematical modeling as they describe relationships between variables. These equations are dubbed "linear" because they create a straight line when graphed.
In the context of our problem, the linear equation, \[ C = 5 + 0.10m \]represents the total cost `C` of Company B's plan depending on minutes used `m`.

• The equation is in the form of `y = mx + c`, where **x** is the variable (minutes in this case) and **c** is the constant (monthly fee).
• The coefficient of `m` (0.10) indicates how much the total cost increases with each additional minute of usage.

Understanding how to set up and interpret these equations offers a powerful means of forecasting costs, managing budgets, and optimizing decisions. Linear equations provide a simple yet accurate method to navigate financial options and make better consumer choices.

Variables in Mathematics

Variables are symbols used in mathematics to represent unknown values or quantities that can change. In mathematical modeling, variables allow us to express dynamic situations succinctly and efficiently.
Considering the problem at hand, `m` represents the variable in Company B's cost equation.

• **Variable `m`**: It stands for the number of minutes used. By assigning a variable, we create a generalized model that applies to any number of minutes.
• **Consistent Representation**: Variables ensure input fluidity, meaning different users with different usage levels can plug their specifics into the equation to determine their costs.

Understanding variables is fundamental in mathematics as they form the core concept behind algebra and calculus. They allow for flexibility and versatility in problem-solving, making the analysis of various scenarios both manageable and insightful. Recognizing their importance is key to mastering mathematical modeling tasks.